The (f,g)-inversion formula and its applications: The (f,g)-summation formula

A characterization of the two functions f(x,y) and g(x,y) in the (f,g)-inversion is presented. As an application to the theory of hypergeometric series, a general bibasic summation formula determined by such two functions f(x,y) and g(x,y) as well as four arbitrary sequences is obtained which unifies the bibasic summation formulas of Gasper and Rahman, Chu, and Macdonald. Furthermore, an alternative proof of the (f,g)-inversion derived from the (f,g)-summation formula is presented. A bilateral (f,g)-inversion containing Schlosser's bilateral matrix inversion as a special case is also obtained.